Analytic solution of the cubic parameter equation for Solution XXIV

Solve analytically the cubic equation governing the ratio of parameters in Solution XXIV of the asymmetric phi^4 equation arising from the n = 1/2 generalized Fokas–Lennels equation, and thereby obtain y in terms of D and m beyond the special case D^2 = 2 and m = 1/4.

Background

For Solution XXIV, the authors derive a cubic equation for y = A/B and simplify it by substituting y = z − D. They state that an analytic solution is unavailable except for one special parameter choice, D^2 = 2 and m = 1/4, where the roots can be written explicitly.

The unresolved problem is to solve this cubic relation analytically for the general admissible parameter range of the periodic kink solution.